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The transverse displacement of a string ...

The transverse displacement of a string (clamped at its both ends) is given by
`y(x, t) = 0.06 sin ((2 x)/(3) x) cos (120 pi t)`
where x and y are in m and t in s. The length of the string is 1.5 m and its mass is `3.0 xx 10^(-2) kg`.
Answer the following :
(a) Does the function represent a travelling wave or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency , and speed of each wave ?
(c ) Determine the tension in the string.

A

It represents a progressive wave of frequency 60 Hz.

B

It represents a stationary wave of frequency 60 Hz.

C

It is the result of superopostion of two waves of wavelength 3m, frequency 60 Hz each travelling with a speed of 180 m/s in opposite direction.

D

Amplitude of this wave is constant.

Text Solution

Verified by Experts

The correct Answer is:
B, C
Comparing ` y = 0.06 sin ((2pi x )/(3)) cos (120 pi t ) ` with `y = a sin (kx) cos (omega t),`
we gets `a = 0.06 m , omega = 120 pi rad//s`
`k = (2pi )/(3) (rad)/(s)`
(a) Given equation does not represent progressive wave as it is not periodic function of `(omega t pm kx). ` Hence, option (A) is false.
(b) `omega =120 pi = 2pi f implies f = 60 Hz`
Option (B) is true.
(c) `v - (omega )/( k) = (120 pi)/((2pi)/(3)) = 180 (m)/(s)` Option (C) is ture.
(d) Given equation represent stationary wave in which different particles perform S.H.M. with different amplitudes. Hence, option (D) is false.
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Knowledge Check

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